## Wednesday, May 18, 2011

### Algebra's Tiling

I don't think algebra tiles or blocks are a panacea for what ails student algebraists, but I do think they are powerful.  Even though as a mathematician I am pretty comfortable with symbolic reasoning, at heart I am a visual thinker.  Having a way to visualize algebra opens up many possibilities for learners, even when they had a good symbolic understanding beforehand.  The goal for me is not to replace symbolic manipulation, but support the concepts.

I designed the activity for students that may have not done a lot of investigation before.  So it starts with a lot of modeling, and then letting them try.  Students did an amazing job.

Even before doing the mental math (making the point that even number operations can be visualized, plus setting the context for the follow up activity), I asked the students to take a  look at the blocks to see what they noticed. Mr. Boeve had had the students play with the blocks the day before, which is an excellent idea.  They noticed corresponding dimensions, colors, different designs.  These were the tiles from Algebra Lab Gear, so there's 1/2 x blocks, 1/4 x blocks, 5 sticks and 25 sticks.

None of the students had seen the visual multiplication before, but they were willing to give it a try.  They made connections as to why the pieces represented what they did.  The verbal connection, x squared, was biggest, but then a few students recognized the x times x relation.  They picked up the symbol to picture representing quickly, and that gave an opportunity to talk about how there are many different ways to write things in math.  They had 2x+3+1x+2 and 2x +3 +x+2 and 3x+5.  We introduced what mathematicians call simplifying, which they connected to fractions.

I raised the problem of negatives - how could we show negatives, because algebra has a lot of those.  They thought we could have two color blocks, or use some kind of design.  How could we do it with the blocks we have?  Maybe we could separate the positives and negatives.  Nice thinking!

Sometimes I think of mini-lessons like these as equipping the students for problems.  The problem list offered practice, light extension and serious problems.  After 15 minutes to do their choice, which they loved having, we came back together and I asked if there were any they wanted to see me do?  They suggested problems, and if there was a student to explain them, they gave it a try.  One of the themes throughout was "give it a go."  I made sure to ask some students who had incomplete or incorrect thinking so we could talk about that, too.  Because the whole situation was different, it helped with making that safe.

In discussing the subtraction problems, they got to three separate ideas: taking away, zero pairs, and adding the opposite.  The students exploring these ideas were able to give reasons for why it made sense with the blocks.

We just worked with the document camera, but if you want virtual algebra tiles, there's the National Virtual Manipulative Library tiles, a two color version from the Michigan Virtual University, and NCTM's Illuminations Algebra Tiles.  None of them are ideal, but all are serviceable.  It's very possible to make homemade algebra tiles, and there was a good article about that in the Mathematics Teacher: "Algebra for All: Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts," Annette Ricks Leitze and Nancy A. Kitt, September 2000, Volume 93, Issue 6.

Thanks to Mr. Boeve and his classes for the nice opportunity!  Jill Beauchamp came along for experience with the algebra blocks, and she was a great support to the kids, so thanks to her, too.

Playing With Blocks

Photo credits: Eamonn @ Flickr